x2 that most factory RVs will not have an inverter, only a converter.
A modern LCD television really does not take much power. YMMV but I have found a cheap modified square wave 400W inverter plugged into the cigarette lighter power plug drives my Furrion flat panel TV just fine.
Just beware of the power consumption which must balance power availability. If your TV consumes 100W... that is 8.3A being pulled from your battery. By way of comparison your heater fan probably pulls half that, so this is a high load for a 12V battery. Especially since it is a constant long duration load. A dead battery can result quicker than you might want unless you offset that consumption with some production.
2015 Rockwood 2604WS (sold as 2016)
2014 Ford F-150 Ecoboost